Balanced-list maximization conjecture for the cd-index

Let $\beta(L)$ denote the beta-value associated with a list $L$. A list is balanced if its entries are either $k$ or $k+1$ for some non-negative integer $k$. Let $L$, $M$, and $L^{\prime}$ be lists. Balanced-list maximization conjecture. There exists a balanced list $B$ having the same degree and length as $M$ such that

$$\beta(L,B,L^{\prime})\geq\beta(L,M,L^{\prime}).$$

Thus, among lists with the prescribed degree and length in the middle position, a balanced list is conjectured to maximize the beta-value in every surrounding context; the source gives no resolution.